Circular Table Seating Arrangement Problem Solution
Learn how to calculate the number of different ways to seat four people around a circular table using the Division Rule in Discrete Math. Understand the concept of distinguishing seating arrangements based on neighbors and how to apply the rule to find the total number of unique seating arrangements.
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Discrete Math: Example 1 of The Division Rule
Example 1 of The Division Rule How many different ways are there to seat four people around a circular table, where two seatings are considered the same when each person has the same left neighbor and the same right neighbor?
Solution We arbitrarily select a seat at the table and label it seat 1. We number the rest of the seats in numerical order, proceeding clockwise around the table. Note that are four ways to select the person for seat 1, three ways to select the person for seat 2, two ways to select the person for seat 3, and one way to select the person for seat 4. Thus, there are 4! = 24 ways to order the given four people for these seats. However, each of the four choices for seat 1 leads to the same arrangement, as we distinguish two arrangements only when one of the people has a different immediate left or immediate right neighbor. Because there are four ways to choose the person for seat 1, by the division rule there are 24/4 = 6 different seating arrangements of four people around the circular table.
References Discrete Mathematics and Its Applications, McGraw-Hill; 7th edition (June 26, 2006). Kenneth Rosen Discrete Mathematics An Open Introduction, 2nd edition. Oscar Levin A Short Course in Discrete Mathematics, 01 Dec 2004, Edward Bender & S. Gill Williamson
